Oriented percolation in one-dimensional beta / |x-y|^2 random cluster model at beta > 1

Details Oriented percolation in one-dimensional beta / |x-y|^2 random cluster model at beta > 1 Oriented percolation in one-dimensional beta / |x-y|^2 random cluster model at beta > 1

2005-06-20

arxiv.org/abs/math/0506404v2

Mathematics

Probability

27 pages, 2 figures, improved version

Scientific paper

We consider the Fortuin-Kasteleyn random cluster model on Z, with edge occupation probabilities p_{} = p if |x-y| = 1, 1 - exp{-beta / |x-y|^2} otherwise, and weighting factor kappa \geq 1. We prove that oriented percolation occurs when beta > 1 and kappa \geq 1 provided p is chosen sufficiently close to 1. In the particular case of the product measure (kappa = 1), this answers a question posed in [IN]. The proof is based on multi-scale analysis.

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